Since the definition of algebraic \(K\) -theory has been extended to many types of structured categories [1]}, the important connection between \(\mathrm {THH}\)  and algebraic \(K\) -theory of ring spectra inspired the definition of similar extensions of (topological) Hochschild homology: to exact categories [2]}, to dg-categories [3]}, and to spectral categories [4]}, as a special case of topological Hochschild homology of bimodules over spectral categories. It can be be challenging to establish properties, such as Morita invariance, of certain of these versions of THH.
